Comma-based lattices: Difference between revisions

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When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a lattice ~~of commas~~ is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma).xlsx|spreadsheet ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png|image]].

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	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a lattice ~~of commas~~ is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

	The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma).xlsx\|spreadsheet ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png\|image]].		The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma).xlsx\|spreadsheet ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png\|image]].